A100324 Square array, read by antidiagonals, where rows are successive self-convolutions of the top row, which equals A003169 shifted one place right.

1, 1, 1, 1, 2, 3, 1, 3, 7, 14, 1, 4, 12, 34, 79, 1, 5, 18, 61, 195, 494, 1, 6, 25, 96, 357, 1230, 3294, 1, 7, 33, 140, 575, 2277, 8246, 22952, 1, 8, 42, 194, 860, 3716, 15372, 57668, 165127, 1, 9, 52, 259, 1224, 5641, 25298, 108018, 415995, 1217270

Offset

0,5

Comments

Column k forms the binomial transform of row k in triangle A100326 for k>=0.

Links

G. C. Greubel, Antidiagonals n = 0..50, flattened

Formula

A(n, k) = Sum_{i=0..k} A(0, k-i)*A(n-1, i) for n>0.

A(0, k) = A003169(k+1) = ( (324*k^2-708*k+360)*A(0, k-1) - (371*k^2-1831*k+2250)*A(0, k-2) +(20*k^2-130*k+210)*A(0, k-3) )/(16*k*(2*k-1)) for k>2, with A(0, 0) = A(0, 1)=1, A(0, 2)=3.

A(n, n) = (n+1)*A032349(n+1).

T(n, k) = A(n-k, k) (Antidiagonal triangle).

T(n, n) = A003169(n+1).

Sum_{k=0..n} T(n, k) = A100325(n) (Antidiagonal row sums).

Example

Array, A(n,k), begins as:
  1, 1,  3,  14,   79,   494,  3294, ...;
  1, 2,  7,  34,  195,  1230,  8246, ...;
  1, 3, 12,  61,  357,  2277, 15372, ...;
  1, 4, 18,  96,  575,  3716, 25298, ...;
  1, 5, 25, 140,  860,  5641, 38775, ...;
  1, 6, 33, 194, 1224,  8160, 56695, ...;
  1, 7, 42, 259, 1680, 11396, 80108, ...;
Antidiagonal triangle, T(n,k), begins as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7,  14;
  1, 4, 12,  34,  79;
  1, 5, 18,  61, 195,  494;
  1, 6, 25,  96, 357, 1230, 3294;
  1, 7, 33, 140, 575, 2277, 8246, 22952;

Mathematica

f[n_]:= f[n]= If[n<2, 1, If[n==2, 3, ((324*n^2-708*n+360)*f[n-1] - (371*n^2-1831*n+2250)*f[n-2] +(20*n^2-130*n+210)*f[n-3])/(16*n*(2*n -1)) ]]; (* f = A003169 *)
A[n_, k_]:= A[n, k]= If[n==0, f[k], If[k==0, 1, Sum[A[0, k-j]*A[n-1, j], {j, 0, k}]]]; (* A = A100324 *)
T[n_, k_]:= A[n-k, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 31 2023 *)

Prog

(PARI) {A(n, k)=if(k==0, 1, if(n>0, sum(i=0, k, A(0, k-i)*A(n-1, i)), if(k==1, 1, if(k==2, 3, ( (324*k^2-708*k+360)*A(0, k-1)-(371*k^2-1831*k+2250)*A(0, k-2)+(20*k^2-130*k+210)*A(0, k-3))/(16*k*(2*k-1)) ))); )}
(SageMath)
def f(n): # f = A003169
    if (n<2): return 1
    elif (n==2): return 3
    else: return ((324*n^2-708*n+360)*f(n-1) - (371*n^2-1831*n+2250)*f(n-2) + (20*n^2-130*n+210)*f(n-3))/(16*n*(2*n-1))
@CachedFunction
def A(n, k): # A = 100324
    if (n==0): return f(k)
    elif (k==0): return 1
    else: return sum( A(0, k-j)*A(n-1, j) for j in range(k+1) )
def T(n, k): return A(n-k, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 31 2023

Crossrefs

Cf. A003169, A032349, A100325, A100326.

Sequence in context: A193092 A263484 A293985 * A121424 A214978 A295380

Adjacent sequences: A100321 A100322 A100323 * A100325 A100326 A100327

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 16 2004

Status

approved